05 July 2017

A math problem solving strategy that's proven to work

As
I said before, there's a tough set of requirements that any problem
solving strategy that's worth teaching to students as a singular
go-to overarching framework must satisfy. I have yet to see a
simple problem solving strategy satisfy all of those requirements,
but here below is something that's been proven to work [1].

The
Formulaic Action Oriented Problem Solving Strategy

It's
"formulaic" because the back-end of the strategy is
centered around using formulas to derive numeric solutions. But
also, this strategy is "formulaic" because it's
step-by-step, bringing students from start to finish of solving a
problem.

It's
"action oriented" because it critically answers for
students the question of "what should I do?"

It's
a "strategy" and not a bag of techniques, and so perhaps
it's better called a "framework". But only because so many
other people call as "strategies" the techniques that one
might use during problem solving.

Certainly
problem solving require techniques, but just as important is learning
terminology.

Terminology
and terms are definitions. Terms have meaning that is denotational.

(1)
diagram
the
problem or situation,
(2)
label
the
diagram,
(3)
choose or write a formula,
(4)
fill the
formula
in with numbers,
(5)
solve for the unknown.

Notice
the first two steps, the front-end of this strategy, depend on
knowing terminology.
The last three steps, the back-end, depend on knowing how to
work with formulas.

Working
with formulas is the most advanced form of
arithmetic
techniques. It is also the most foundational form of algebraic
techniques. Therefore knowing what to do with formulas is an
essential skill, and set of techniques, to learn: especially to
bridge students from arithmetic to algebra.

Thus
this strategy's back-end could be equivalently used with arithmetic,
formula, or algebra based math, depending on the student's
mathematical maturity. Also equivalently, though it ought to be
rarely (especially in grade school), the back-end here could be
swapped out to use algorithmic based math as well --- and I really
mean algorithmic as defined technically in computing science, not
just any step-by-step non-mathematical short-cut that someone without
higher mathematical maturity concocts.

The
formula
centrein that problem
solving strategy's back-end

The
formula centre consists of the last three steps. Recall they are:

(3)
choose or write an appropriate formula
(4)
Then filling into the formula the appropriate numbers to replace the
variables in that formula.
(5)
And finally, appropriately solving for the unknown variable.

The
simpler the formula, the more it involves just arithmetic skills:
including the arithmetic technique of number substitution
which should've been previously learned in junior high school, and
similarly also the arithmetic technique of balancing equations
which is often confused or misunderstood as algebra.

The
more complex the formula, the more it needs algebraic techniques:
especially as you need better knowledge of the algebraic
technique ofcancellation for
canceling
terms, factors, and fractions until getting the formula into the
desired form to
solve
for the unknown.

This
problem solving technique works broadly, e.g. from topics like
measurement (unit conversion, areas and volume), to
trigonometry.

It
even works for topics like roots and powers --- but
then that involves some more advanced skills like the algebraic
technique of formula substitution:
replacing a whole, or part, of an equation with another expression,
kind of like instead of filling an equation in with numbers, you fill
it in with expressions.

It
also works for Factors and products --- but students
would need to learn either the abstract area model or algebra
for things like distribution, factoring out GCD, binomial
decomposition, etc. But all these are just applications of the
basic algebraic techniques used in the Formulaic Action Oriented
Problem Solving Strategy!

It
also works for problem solving with Relations and functions.
Of course, functions are just special formulas, and Graph
Sketching and Interpretation techniques would be needed.

Similarly,
Linear functions. And transforming between the 3
forms of linear equations with algebra mainly requires the algebraic
technique of cancellation: cancelling terms, factors, and
fractions until getting desired form.

Also
Linear systems. Linear Systems can be solved
using Graph Sketching and Interpretation techniques, and also
with algebra (algebraic techniques of: formula
substitution, and formulaelimination).

Let's
see: that's just 3 foundational algebraic techniques and two visual
(diagrammatic, graphical) techniques! All working within one
problem solving strategy.

And
the many variations of those.
And
a lot of terminology!

Turns
out Math isn't that hard: the above covers a large number of basic
fields of math that students have to learn, and we counted up just 5
foundational techniques and one problem solving strategy.

[1]
I would know because I've seen it used by some very strong and some
very weak math students, all to great success! In fact, the
above was written in around 2014 August after such observations.